ICTS

The goal of this course is to introduce the basic tools of microlocal analysis and to show how these techniques can be used for the study of wave control. We start with the wave front set and pseudo-differential calculus. Then, we present the propagation of singularities theorem of L.H ̈ormander. Finally, we present the notion of microlocal defect measures with their microlocal properties ( elliptic regularity and support propagation ). All these facts will be illustrated by various applications, in particular the observability of the wave equation.

We introduce in an abstract framework the notion of infinite dimensional time invariant well posed linear system (with inputs and outputs). We provide examples of such systems described by some classical evolution PDEs with distributed control and observation. We next define the fundamental notion of reachable space for this type of systems and we relate its properties with various controllability and observability types. We illustrate this theory by studying the reachability and controllability properties examples of various systems described by linear evolution partial differential equations.

In this lecture, I will present Carleman Estimates for the Laplace operator and some of its applications. In particular, I will discuss the Cauchy problem for the Laplace operator, its uniqueness and stability properties, and then introduce general Carleman Estimates for the Laplace operator. As an application, I will give a quantification of the unique continuation property for the Laplace operator with respect to a potential, and discuss the link with the Landis conjecture, and applications to the optimal control problem for a semilinear elliptic PDE.

We introduce in an abstract framework the notion of infinite dimensional time invariant well posed linear system (with inputs and outputs). We provide examples of such systems described by some classical evolution PDEs with distributed control and observation. We next define the fundamental notion of reachable space for this type of systems and we relate its properties with various controllability and observability types. We illustrate this theory by studying the reachability and controllability properties examples of various systems described by linear evolution partial differential equations.

In this lecture, I will present Carleman Estimates for the Laplace operator and some of its applications. In particular, I will discuss the Cauchy problem for the Laplace operator, its uniqueness and stability properties, and then introduce general Carleman Estimates for the Laplace operator. As an application, I will give a quantification of the unique continuation property for the Laplace operator with respect to a potential, and discuss the link with the Landis conjecture, and applications to the optimal control problem for a semilinear elliptic PDE.

We introduce in an abstract framework the notion of infinite dimensional time invariant well posed linear system (with inputs and outputs). We provide examples of such systems described by some classical evolution PDEs with distributed control and observation. We next define the fundamental notion of reachable space for this type of systems and we relate its properties with various controllability and observability types. We illustrate this theory by studying the reachability and controllability properties examples of various systems described by linear evolution partial differential equations.

A control system is differentially flat if every trajectory can be expressed in terms of a function (called a flat output) and its derivatives. The flatness approach, introduced by Michel Fliess, Jean L ́evine, Philippe Martin and Pierre Rouchon in 1995 for ODEs, was next applied to PDEs by B ́eatrice Laroche, Philippe Martin and Pierre Rouchon in 2000 to derive the approximate controllability of the heat equation.

In this course, we will show how the flatness approach can be used to derive null or exact controllability results for many evolution equations with some boundary controls.

We first show how to obtain the null controllability of the 1D heat equation, or of the ND heat equation in cylinders, providing both the control input and the trajectory as series. We shall see how to compute easily a lot of derivatives of the flat output to compute the partial sums in the series. Next, we show how to apply the flatness approach to the linear 1D Schr ̈odinger equation and to the linear Korteweg-de Vries equation.

We proceed to the null controllability of the heat equation with variable coefficients, namely

\[ (a(x)u_x)_x+ b(x)u_x+c(x)u=\rho (x)u_t.\]

Here the coefficients a, b, c, ρ are merely assumed to be measurable, and we stress that Carleman estimates (if any) are not yet available. We mainly assume first that a ∈ L¹ and 1/a ∈ L¹ (weak degeneracy). Next, we pass to the case of a strong degeneracy at one point of the boundary. The flatness approach is combined with Pr ̈ufer method to obtain the asymptotics of the eigenvalues. Next, we are concerned with the determination of the reachable states of the heat equation, first for the boundary control, and next for the distributed control.
Finally, we address the issue of the exact controllability of a semilinear heat equation, and show how a Cauchy problem in the variable x can be solved by using scales of Banach spaces constituted of Gevrey functions. We obtain a local exact controllability in a space of holomorphic functions. The method is next extended to any anisotropic 1D equation of the form

\[ \partial _t^N y=\zeta _M \partial _x^M y +f(x,y,...,\partial _x^{M-1}y), \quad 1\le N<M,\]

that may be severely ill-posed. We obtain again some local exact controllability result in some space of holomorphic functions. The result can be applied to

1. the Korteweg-de Vries equation: $\partial _t y+\partial _x^3 y + \partial _xy+y\partial _xy=0$;

2. the ``good'' and ``bad'' Boussinesq equation: $\partial _t^2 y =\pm \partial_x^4 y +\partial _x ^2 y -\partial _x^2 (y^2)$;

3. the Ginzburg-Landau equation: $\partial _t y = e^{i\theta} \partial _x ^2 y + e^{i\varphi} |y|^2y$ where $\theta,\varphi\in\R$;

4. the Kuramoto-Sivashinsky equation: $\partial _ty +\partial _x ^4 y +\partial _x^2 y +y\partial _x y=0$.

Coupled systems appear everywhere in complex models and in some cases there are different time scales involved. The coupling and the scales make this kind of system very difficult to study from theoretical and computational viewpoints. One hopes that some particular properties of the system could be studied through simpler uncoupled systems. This is what the singular perturbation method (SPM) does concerning stability properties. The SPM approach has been introduced for ordinary differential equations and can also be applied for partial differential equations but in the latter case there are no general theorems and stability properties have to be obtained for each particular system. In this course we introduce the SPM for infinite-dimensional systems and obtain stability results. We will consider parabolic, hyperbolic and dispersive equations appearing in coupled systems in some cases also involving ordinary differential equations.

In this talk I will present some results for a nonlocal Cahn-Hilliard-Brinkman system, which models phase separation of binary fluids in porous media. To derive the first order optimality conditions one needs to prove differentiability of the control to state map which in turn requires higher regularity of the solutions. We will consider the cases of regular as well as singular potential. In both cases we study the tracking type optimal control problem with control in the velocity equation. The results like the existence of optimal control and characterization of optimal control in terms of the solution of the corresponding adjoint system will be discussed.

Coupled systems appear everywhere in complex models and in some cases there are different time scales involved. The coupling and the scales make this kind of system very difficult to study from theoretical and computational viewpoints. One hopes that some particular properties of the system could be studied through simpler uncoupled systems. This is what the singular perturbation method (SPM) does concerning stability properties. The SPM approach has been introduced for ordinary differential equations and can also be applied for partial differential equations but in the latter case there are no general theorems and stability properties have to be obtained for each particular system. In this course we introduce the SPM for infinite-dimensional systems and obtain stability results. We will consider parabolic, hyperbolic and dispersive equations appearing in coupled systems in some cases also involving ordinary differential equations.

In this talk, we discuss the pointwise control constrained Dirichlet boundary optimal control problem governed by an elliptic problem and its numerical approximation by finite element methods.

The martingale problem for degenerate diffusions in $\mathcal{R}^{d}$ with continuous coefficients may not have unique solution measures. However, the solution set $A_{x}$ of solution measures for any initial condition $x \in \mathcal{R}^{d}$ is compact convex and non-empty. The Markov selection problem is to select a $P_{x} \in A_{x}$ for each $x$ such that the family $\{P_{x} \}$ satisfies the Chapman- Kolmogorov equations, thus forming a Markov process. The Krylov selection procedure is a well known resolution of this problem. In this talk, I shall sketch an alternative approach based on small noise limits. The talk will include an overview of diffusion theory. (Joint work with K. Suresh Kumar and Anugu Sumith Reddy).

These lectures are mainly based on different joint works with Mehdi Badra.

We consider systems of the form

$$z' = Az + Bf,\quad z(0)=z_0,$$

where A is the infinitesimal generator of an analytic semigroup on a Hilbert space Z, B is a control operator, f ∈ U is the control variable, and U is another Hilbert space. We consider approximate systems

$$z_\varepsilon' = A_\varepsilon z_\varepsilon + B_\varepsilon f,\quad z_\varepsilon(0)=z_{0,\varepsilon},$$

where $A_\varepsilon$ is the infinitesimal generator of an analytic semigroup on a Hilbert space $Z_\varepsilon$, $B_\varepsilon$ is a control operator. For numerical approximations by Finite Element Methods, $\varepsilon =h$ is the mesh size of the triangulation. For approximations by a penalty method, $\varepsilon $ is the penalty parameter.

We are interested in finding feedback gains $K\in{\mathcal L}(Z,U)$ such that $(e^{t(A+BK)})_{t\geq 0}$ is exponentially stable on $Z$. We would like to know if such  feedback gains  can be approximated by

feedback gains $K_\varepsilon\in{\mathcal L}(Z_\varepsilon,U)$, where $K_\varepsilon$ is chosen so that

$(e^{t(A_\varepsilon+B_\varepsilon K_\varepsilon)})_{t\geq 0}$ is exponentially stable on $Z_\varepsilon$.

We consider the case of nonconforming approximations, that is the case where $Z_\varepsilon \not \subset Z$. We assume that

$Z  \subset H$, $Z_\varepsilon \subset H$, $H$ is a Hilbert space, and that there exist projectors $P\in{\mathcal L}(H)$ and

$P_\varepsilon\in{\mathcal L}(H)$ such that $P H = Z$ and $P_\varepsilon H = Z_\varepsilon$.

In Lecture 1, we give sufficient conditions on the triplets $(A,B,P)$ and $(A_\varepsilon,B_\varepsilon,P_\varepsilon)$ and some approximations assumptions with which we prove that Riccati based feedback gains for $(A,B)$ can be approximated by

feedback gains for $(A_\varepsilon,B_\varepsilon)$.

In Lecture 2, we show that the stabilization of the Oseen system (the linearized Navier-Stokes equations around an unstable stationary solution) by a boundary control, and its approximation by a F.E.M. fits into the functional setting introduced in Lecture 1.

 We obtain new error estimates for the F.E. approximation of the stationary Oseen system with nonhomogeneous boundary conditions, in convex or non-convex polyhedral domains in ${\mathbb R}^3$.

In Lecture 3, we improve the convergence rates obtained in Lecture 1, by using Reduced Order Model based on spectral projections. We give some applications.

In Lecture 4, we study Fluid-Structure-Interaction systems in the case where the structure is a damped elastic beam or shell located at the boundary of the fluid domain. We recall some existence results of maximal in time strong solutions to these systems. We study the local stabilization of these systems around unstable stationary solutions. We show that the feedback gains can be determined by the method introduced in Lecture 3. Some hints will be given on the numerical approximation of a linearized model.

This course follows the lectures given by B. Dehman. We will begin by discussing the relationships between controllability, observability and exponential stability (Haraux theorem). Then we will study observability of wave equations on measurable observation sets. Finally, we will try to determine the best possible observation set, that is, we will study the optimal observability problem, for wave equations but also for parabolic equations for which results are strongly different.

This course focuses on small-time local controllability (STLC) for affine control systems with scalar input x0 = f0(x) + u(t)f1(x). First, we will look at linear theory (Kalman rank condition), STLC proofs by linearization and by power series expansion, and their links with linear or Holderian cost estimates. Going further, we will introduce Lie brackets of vector fields and state the Lie algebra rank condition, which is a necessary condition for STLC (but not sufficient, when drift is active). Thanks to a new representation formula for the state, we will demonstrate STLC necessary conditions, including the most classical ones (Sussmann, Stefani). This will show the crucial role of interpolation inequalities (e.g. Gagliardo-Nirenberg) in the proof of STLC necessary conditions. Finally, using the emblematic example of the 1D Schroodinger equation with bilinear control, we will see how these results can be extended to PDEs.

These lectures are mainly based on different joint works with Mehdi Badra.

We consider systems of the form

$$z' = Az + Bf,\quad z(0)=z_0,$$

where A is the infinitesimal generator of an analytic semigroup on a Hilbert space Z, B is a control operator, f ∈ U is the control variable, and U is another Hilbert space. We consider approximate systems

$$z_\varepsilon' = A_\varepsilon z_\varepsilon + B_\varepsilon f,\quad z_\varepsilon(0)=z_{0,\varepsilon},$$

where $A_\varepsilon$ is the infinitesimal generator of an analytic semigroup on a Hilbert space $Z_\varepsilon$, $B_\varepsilon$ is a control operator. For numerical approximations by Finite Element Methods, $\varepsilon =h$ is the mesh size of the triangulation. For approximations by a penalty method, $\varepsilon $ is the penalty parameter.

We are interested in finding feedback gains $K\in{\mathcal L}(Z,U)$ such that $(e^{t(A+BK)})_{t\geq 0}$ is exponentially stable on $Z$. We would like to know if such  feedback gains  can be approximated by

feedback gains $K_\varepsilon\in{\mathcal L}(Z_\varepsilon,U)$, where $K_\varepsilon$ is chosen so that

$(e^{t(A_\varepsilon+B_\varepsilon K_\varepsilon)})_{t\geq 0}$ is exponentially stable on $Z_\varepsilon$.

We consider the case of nonconforming approximations, that is the case where $Z_\varepsilon \not \subset Z$. We assume that

$Z  \subset H$, $Z_\varepsilon \subset H$, $H$ is a Hilbert space, and that there exist projectors $P\in{\mathcal L}(H)$ and

$P_\varepsilon\in{\mathcal L}(H)$ such that $P H = Z$ and $P_\varepsilon H = Z_\varepsilon$.

In Lecture 1, we give sufficient conditions on the triplets $(A,B,P)$ and $(A_\varepsilon,B_\varepsilon,P_\varepsilon)$ and some approximations assumptions with which we prove that Riccati based feedback gains for $(A,B)$ can be approximated by

feedback gains for $(A_\varepsilon,B_\varepsilon)$.

In Lecture 2, we show that the stabilization of the Oseen system (the linearized Navier-Stokes equations around an unstable stationary solution) by a boundary control, and its approximation by a F.E.M. fits into the functional setting introduced in Lecture 1.

 We obtain new error estimates for the F.E. approximation of the stationary Oseen system with nonhomogeneous boundary conditions, in convex or non-convex polyhedral domains in ${\mathbb R}^3$.

In Lecture 3, we improve the convergence rates obtained in Lecture 1, by using Reduced Order Model based on spectral projections. We give some applications.

In Lecture 4, we study Fluid-Structure-Interaction systems in the case where the structure is a damped elastic beam or shell located at the boundary of the fluid domain. We recall some existence results of maximal in time strong solutions to these systems. We study the local stabilization of these systems around unstable stationary solutions. We show that the feedback gains can be determined by the method introduced in Lecture 3. Some hints will be given on the numerical approximation of a linearized model.

This course follows the lectures given by B. Dehman. We will begin by discussing the relationships between controllability, observability and exponential stability (Haraux theorem). Then we will study observability of wave equations on measurable observation sets. Finally, we will try to determine the best possible observation set, that is, we will study the optimal observability problem, for wave equations but also for parabolic equations for which results are strongly different.

This course focuses on small-time local controllability (STLC) for affine control systems with scalar input $x'=f_{0}(x)+u(t)f_{1}(x).$. First, we will look at linear theory (Kalman rank condition), STLC proofs by linearization and by power series expansion, and their links with linear or Holderian cost estimates. Going further, we will introduce Lie brackets of vector fields and state the Lie algebra rank condition, which is a necessary condition for STLC (but not sufficient, when drift is active). Thanks to a new representation formula for the state, we will demonstrate STLC necessary conditions, including the most classical ones (Sussmann, Stefani). This will show the crucial role of interpolation inequalities (e.g. Gagliardo-Nirenberg) in the proof of STLC necessary conditions. Finally, using the emblematic example of the 1D Schroodinger equation with bilinear control, we will see how these results can be extended to PDEs.

These lectures are mainly based on different joint works with Mehdi Badra.

We consider systems of the form

$$z' = Az + Bf,\quad z(0)=z_0,$$

where A is the infinitesimal generator of an analytic semigroup on a Hilbert space Z, B is a control operator, f ∈ U is the control variable, and U is another Hilbert space. We consider approximate systems

$$z_\varepsilon' = A_\varepsilon z_\varepsilon + B_\varepsilon f,\quad z_\varepsilon(0)=z_{0,\varepsilon},$$

where $A_\varepsilon$ is the infinitesimal generator of an analytic semigroup on a Hilbert space $Z_\varepsilon$, $B_\varepsilon$ is a control operator. For numerical approximations by Finite Element Methods, $\varepsilon =h$ is the mesh size of the triangulation. For approximations by a penalty method, $\varepsilon $ is the penalty parameter.

We are interested in finding feedback gains $K\in{\mathcal L}(Z,U)$ such that $(e^{t(A+BK)})_{t\geq 0}$ is exponentially stable on $Z$. We would like to know if such  feedback gains  can be approximated by

feedback gains $K_\varepsilon\in{\mathcal L}(Z_\varepsilon,U)$, where $K_\varepsilon$ is chosen so that

$(e^{t(A_\varepsilon+B_\varepsilon K_\varepsilon)})_{t\geq 0}$ is exponentially stable on $Z_\varepsilon$.

We consider the case of nonconforming approximations, that is the case where $Z_\varepsilon \not \subset Z$. We assume that

$Z  \subset H$, $Z_\varepsilon \subset H$, $H$ is a Hilbert space, and that there exist projectors $P\in{\mathcal L}(H)$ and

$P_\varepsilon\in{\mathcal L}(H)$ such that $P H = Z$ and $P_\varepsilon H = Z_\varepsilon$.

In Lecture 1, we give sufficient conditions on the triplets $(A,B,P)$ and $(A_\varepsilon,B_\varepsilon,P_\varepsilon)$ and some approximations assumptions with which we prove that Riccati based feedback gains for $(A,B)$ can be approximated by

feedback gains for $(A_\varepsilon,B_\varepsilon)$.

In Lecture 2, we show that the stabilization of the Oseen system (the linearized Navier-Stokes equations around an unstable stationary solution) by a boundary control, and its approximation by a F.E.M. fits into the functional setting introduced in Lecture 1.

 We obtain new error estimates for the F.E. approximation of the stationary Oseen system with nonhomogeneous boundary conditions, in convex or non-convex polyhedral domains in ${\mathbb R}^3$.

In Lecture 3, we improve the convergence rates obtained in Lecture 1, by using Reduced Order Model based on spectral projections. We give some applications.

In Lecture 4, we study Fluid-Structure-Interaction systems in the case where the structure is a damped elastic beam or shell located at the boundary of the fluid domain. We recall some existence results of maximal in time strong solutions to these systems. We study the local stabilization of these systems around unstable stationary solutions. We show that the feedback gains can be determined by the method introduced in Lecture 3. Some hints will be given on the numerical approximation of a linearized model.

A control system is differentially flat if every trajectory can be expressed in terms of a function (called a flat output) and its derivatives. The flatness approach, introduced by Michel Fliess, Jean L ́evine, Philippe Martin and Pierre Rouchon in 1995 for ODEs, was next applied to PDEs by B ́eatrice Laroche, Philippe Martin and Pierre Rouchon in 2000 to derive the approximate controllability of the heat equation.

In this course, we will show how the flatness approach can be used to derive null or exact controllability results for many evolution equations with some boundary controls.

We first show how to obtain the null controllability of the 1D heat equation, or of the ND heat equation in cylinders, providing both the control input and the trajectory as series. We shall see how to compute easily a lot of derivatives of the flat output to compute the partial sums in the series. Next, we show how to apply the flatness approach to the linear 1D Schr ̈odinger equation and to the linear Korteweg-de Vries equation.

We proceed to the null controllability of the heat equation with variable coefficients, namely

\[ (a(x)u_x)_x+ b(x)u_x+c(x)u=\rho (x)u_t.\]

Here the coefficients a, b, c, ρ are merely assumed to be measurable, and we stress that Carleman estimates (if any) are not yet available. We mainly assume first that a ∈ L¹ and 1/a ∈ L¹ (weak degeneracy). Next, we pass to the case of a strong degeneracy at one point of the boundary. The flatness approach is combined with Pr ̈ufer method to obtain the asymptotics of the eigenvalues. Next, we are concerned with the determination of the reachable states of the heat equation, first for the boundary control, and next for the distributed control.
Finally, we address the issue of the exact controllability of a semilinear heat equation, and show how a Cauchy problem in the variable x can be solved by using scales of Banach spaces constituted of Gevrey functions. We obtain a local exact controllability in a space of holomorphic functions. The method is next extended to any anisotropic 1D equation of the form

\[ \partial _t^N y=\zeta _M \partial _x^M y +f(x,y,...,\partial _x^{M-1}y), \quad 1\le N<M,\]

that may be severely ill-posed. We obtain again some local exact controllability result in some space of holomorphic functions. The result can be applied to

1. the Korteweg-de Vries equation: $\partial _t y+\partial _x^3 y + \partial _xy+y\partial _xy=0$;

2. the ``good'' and ``bad'' Boussinesq equation: $\partial _t^2 y =\pm \partial_x^4 y +\partial _x ^2 y -\partial _x^2 (y^2)$;

3. the Ginzburg-Landau equation: $\partial _t y = e^{i\theta} \partial _x ^2 y + e^{i\varphi} |y|^2y$ where $\theta,\varphi\in\R$;

4. the Kuramoto-Sivashinsky equation: $\partial _ty +\partial _x ^4 y +\partial _x^2 y +y\partial _x y=0$.

These lectures are mainly based on different joint works with Mehdi Badra.

We consider systems of the form

$$z' = Az + Bf,\quad z(0)=z_0,$$

where A is the infinitesimal generator of an analytic semigroup on a Hilbert space Z, B is a control operator, f ∈ U is the control variable, and U is another Hilbert space. We consider approximate systems

$$z_\varepsilon' = A_\varepsilon z_\varepsilon + B_\varepsilon f,\quad z_\varepsilon(0)=z_{0,\varepsilon},$$

where $A_\varepsilon$ is the infinitesimal generator of an analytic semigroup on a Hilbert space $Z_\varepsilon$, $B_\varepsilon$ is a control operator. For numerical approximations by Finite Element Methods, $\varepsilon =h$ is the mesh size of the triangulation. For approximations by a penalty method, $\varepsilon $ is the penalty parameter.

We are interested in finding feedback gains $K\in{\mathcal L}(Z,U)$ such that $(e^{t(A+BK)})_{t\geq 0}$ is exponentially stable on $Z$. We would like to know if such  feedback gains  can be approximated by

feedback gains $K_\varepsilon\in{\mathcal L}(Z_\varepsilon,U)$, where $K_\varepsilon$ is chosen so that

$(e^{t(A_\varepsilon+B_\varepsilon K_\varepsilon)})_{t\geq 0}$ is exponentially stable on $Z_\varepsilon$.

We consider the case of nonconforming approximations, that is the case where $Z_\varepsilon \not \subset Z$. We assume that

$Z  \subset H$, $Z_\varepsilon \subset H$, $H$ is a Hilbert space, and that there exist projectors $P\in{\mathcal L}(H)$ and

$P_\varepsilon\in{\mathcal L}(H)$ such that $P H = Z$ and $P_\varepsilon H = Z_\varepsilon$.

In Lecture 1, we give sufficient conditions on the triplets $(A,B,P)$ and $(A_\varepsilon,B_\varepsilon,P_\varepsilon)$ and some approximations assumptions with which we prove that Riccati based feedback gains for $(A,B)$ can be approximated by

feedback gains for $(A_\varepsilon,B_\varepsilon)$.

In Lecture 2, we show that the stabilization of the Oseen system (the linearized Navier-Stokes equations around an unstable stationary solution) by a boundary control, and its approximation by a F.E.M. fits into the functional setting introduced in Lecture 1.

 We obtain new error estimates for the F.E. approximation of the stationary Oseen system with nonhomogeneous boundary conditions, in convex or non-convex polyhedral domains in ${\mathbb R}^3$.

In Lecture 3, we improve the convergence rates obtained in Lecture 1, by using Reduced Order Model based on spectral projections. We give some applications.

In Lecture 4, we study Fluid-Structure-Interaction systems in the case where the structure is a damped elastic beam or shell located at the boundary of the fluid domain. We recall some existence results of maximal in time strong solutions to these systems. We study the local stabilization of these systems around unstable stationary solutions. We show that the feedback gains can be determined by the method introduced in Lecture 3. Some hints will be given on the numerical approximation of a linearized model.